What is the role of logic in Mathematics?

Nov 27, 2017
• Aidan Rocke

Introduction:

The orthodox belief among pure mathematicians is that the foundations of mathematics are grounded in a few sacred axioms
and set theory where logic naturally has a central role in its development. However, by means of a simple thought experiment
I show that curiosity, more than logic, is essential for the development of mathematics. Moreover, I argue that
curiosity is firmly grounded in both our sensorimotor experience and the tools we use for doing mathematics.

This leads to a holistic account of the foundations of mathematics which challenges the Platonic notion that
‘pure’ mathematics is discovered and makes the case that the envelope of potential mathematical
discoveries is parametrised by both human morphology and technologies for doing mathematics. Crucially, this ‘Cyborg’ view
of mathematics has important implications for investigations on the foundations of mathematics as well as the manner
mathematics is taught at the university level.

The role of logic in mathematics:

While the importance of axiomatics and set theory in structuring mathematics is undeniable, I think we should not lose sight
of what logic actually provides:

A system for verifying our discoveries to an axiomatic level of detail.

A method for communicating our mathematical discoveries in a convincing manner.

In truth, the second argument has much greater weight than the first since an important consequence of Gödel’s incompleteness
theorems is that logic doesn’t guarantee the permanence of our mathematical discoveries. Furthermore, very few mathematicians
use formal proof assistants like Coq or Isabelle to write their mathematical proofs although proof assistants are practically
essential for verification at an axiomatic level of detail. How can we explain this?

Like all humans, mathematicians pursue rigor only to the extent that its cost justifies the reward. That said, if logical verification
isn’t essential to mathematics what could possibly be the vital force behind its development?

The importance of curiosity:

While I would grant that logical verification is important for problem solving in mathematics, if mathematics was reducible to
problem solving we would have no more than one mathematical question to answer(ex. 2+2=?) and there wouldn’t have been a field
of mathematics. In other words, there has to be some intrinsic motivation in all mathematicians which drives them to not only
solve problems but also seek out problems to solve. From this it follows that intrinsic motivation(or curiosity) has a much greater
role than logic in explaining why there are multiple branches of mathematics. In fact, this implies that curiosity not logic has to
be the vital force which guides its development.

Such a line of reasoning is especially relevant to investigations on the foundations of mathematics as it immediately raises doubts
on the platonic account of mathematics. This however raises important epistemological questions concerning the nature of curiosity.

The origin and development of mathematics:

In [2], Poincaré famously argues that primitive mathematical notions like size, continuity and number have imprecise perceptual origins. A child can learn to tell the difference in size between a big dog and a small dog without having to first learn about the greater than relation. Such perceptual faculties effectively serve as good priors for learning mathematics, a task which would be considerably harder otherwise. In addition, there is a wide range of scientific evidence presented in [1] demonstrating that-besides being the origin of our mathematical knowledge-our sensorimotor experience is an essential guide in our mathematical development. This means that our curiosity is constrained by both our morphology and the tools we use for doing mathematics.

While mathematical reasoning often conforms to mathematical principles, it is typically implemented in a sensorimotor loop which includes a device for data-input(ex. pen/pencil) and material for data-storage(ex. paper). In this context, the authors of [1] advance a Cyborg view of mathematics:

…the active manipulation of physical notations plays the role of ‘guiding’ the biological machinery through an abstract mathematical problem space-one that may exceed the space of otherwise solveable problems.

Although many mathematicians might contest this, I wonder whether any mathematician can do advanced mathematics without pen and paper, or a functional substitute. We must also acknowledge the increasingly important role of the computer for doing research-level mathematics.

In addition, we must note a more subtle but equally significant technology; mathematical notation has evolved over time by a process which isn’t arbitrary. While the space of satisfactory mathematical notations might be large, most randomly generated notations are bad for doing mathematics which is why mathematicians define rules of thumb for good notation. The triumph of Leibniz notation over Newton’s notation is a concrete example of this. Moreover, Terrence Tao once wrote a full blog post on this issue which includes the following quote due to Alfred North Whitehead:

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race.

Yet, this statement flies in the face of Cognitive Science orthodoxy as stated in [1]:

Cognitive scientists have traditionally viewed this capacity-the capacity for symbolic reasoning-as grounded in the ability to internally represent numbers, logical relationships, and mathematical rules in an abstract, amodal fashion.

Clearly, this line of reasoning is absurd. If anything both scientific and empirical evidence strongly indicates that our sensorimotor experience is an essential substrate for mathematical thought and not merely a translational medium. When combined with the importance of curiosity it follows that we
have to encourage individual experimentation with technologies aiding mathematical activity in order to maximise the collective human potential for
mathematical discovery.

Conclusion:

Having laid out these arguments, I think it’s clear that the Cyborg view of mathematics provides more stable foundations for mathematics than the orthodox view which is not only scientifically and empirically baseless, but also diminishes our collective potential for mathematical discovery. In particular, I would like to point out a few key innovations in the Cyborg tradition which have yet to be fully appreciated at the university level.

The first is the use of online blogs for communicating mathematical ideas as written homework/projects can be very isolating rather than engaging. You generally get very little feedback even if you do get a good mark which trivialises the activity. Second, is the creation of Polymath projects for exploring the role of large-scale self-organizing collaboration among students. Finally, I think mathematicians of all levels of ability can benefit from using Jupyter notebooks for interactive experimental mathematics as I have whenever investigating problems in combinatorics or probability.

In my opinion, these innovations indicate yet-unrealised potential. Indeed, I believe that if the majority of mathematicians transition towards a Cyborg perspective of mathematical foundations, we shall witness a much more creative period of mathematics.